what light? : or the question of parallel pumping in...

WHAT LIGHT?

or

THE QUESTION OF PARALLEL PUMPING

IN FERROMAGNETS

Andrew Marcus Lear Kurn

B.A., Reed College, 1970

A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF

THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

in the Department

of

Physics

SIMON FRASER UNIVERSITY

April 1977

No copyright reserved.

APPROVAL

Name: Andrew Marcus Lear Kurn

Degree: Master of Science

Thesis T i t l e : What Light? o r The Question of P a r a l l e l Pumping i n Ferromagnets

Examining Committee:

Chairman: Suso Gygax

M. F. Cochran Senior Supervisor

E. D. Crozier

B. L. Jones

B;-etislav ~ e i n r i c v h Universi ty Examiner Research Associate

Simon Fraser Universi ty

PARTIAL COPYRIGHT LICENSE

I hereby g r a n t t o Simon F r a s e r U n i v e r s i t y t h e r i g h t t o lend

my t h e s i s o r d i s s e r t a t i o n ( t h e t i t l e of which i s shown below) t o u s e r e

of t h e Simon F r a s e r U n i v e r s i t y L i b r a r y , and t o make p a r t i a l o r s i n g l e

c o p i e s o n l y f o r . s u c h u s e r s o r In response t o a r e q u e s t from t h e l i b r a r y

of a n y o t h e r u n i v e r s i t y , o r o t h e r e d u c a t i o n a l i n s t i t u t i o n , on i t s 'own

b e h a l f o r f o r one of i t s u s e r s . I f u r t h e r a g r e e t h a t pe rmiss ion f o r

m u l t i p l e copying of t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d

by me o r t h e Dean of Graduate S t u d i e s . It i s unders tood t h a t copying

o r p u b l i c a t i o n of t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t b e a l lowed

w i t h o u t my w r i t t e n pe rmiss ion .

T i t l e of T h e s i s / ~ i s s e r t a t i o n :

V h a t L i g h t ? o r T h e Q u e s t i o n o f

P a r a l l e l P u m p i n g i n F e r r o m a g n e t s .

Author :

( s i g n a t u r e )

A n d r e w M . K U R N

(name )

( d a t e )

iii

ABSTRACT

A b r i e f i n t r o d u c t i o n i s given t o t h e response of a ferromagnet ic

p l a t e t o monochromatic microwave r a d i a t i o n when i t i s s u b j e c t t o a

cons tan t magnetic f i e l d o r i en t ed p a r a l l e l t o t he p l a t e s u r f a c e . A

theory d e s c r i b i n g t h i s response is presented which has a l i n e a r dependence

on the amplitude of t h e i n c i d e n t r a d i a t i o n . The theory of Lieu and

Alexandrakis f o r the case i n which the microwave and s t a t i c magnetic

f i e l d s a r e p a r a l l e l , which has a non-l inear dependence on the ampli tude,

is shown t o be i n e r r o r .

Microwave t ransmiss ion through a 40um t h i c k Supermalloy f o i l

(d l6 = 22.5) w i th RF and s t a t i c magnetic f i e l d s p a r a l l e l revea led

a d i s t o r t e d r e p l i c a of t he perpendicular t ransmiss ion s i g n a l , b u t

a t t e n u a t e d 600-fold i n ampli tude. Transmission through a 7.2um t h i c k

f o i l of n i c k e l gave a f i e l d independent s i g n a 1 a t f i e l d s g r e a t e r than

approximately 1 koe i n agreement w i th t h e obse rva t ions r epo r t ed by

Lieu and Alexandrakis. This background s i g n a l w a s found t o be c o n s i s t e n t

i n s t r e n g t h wi th t h a t p red ic t ed by t h e theory descr ibed he re . For

both f o i l s an e x t r a t ransmiss ion s i g n a l due t o ferromagnet ic domain

w a l l s was observed near ze ro s t a t i c f i e l d .

But, s o f t ! What l i g h t through yonder window breaks? ~ & e o and Juliet 11.2.1

I wish t o extend my g r a t i t u d e t o S tephanie Cochran

who provided t h e g e r b i l s , t o Doreen and Marion and Rick

who f ed t h e g e r b i l s , t o Maureen MCIlmoyl f o r b r u t e s t r e n g t h ,

t o Wendell Bishop f o r a kind of low animal cunning,

t o John Cochran who provided advice , bo th good and

bad, i n h e r o i c q u a n t i t y , and t o God who c r e a t e d

a l l t h e above and invented magnetism t o boot : t h e

knight i n H i s chessgame.

I am indebted t o t h e members of ou r r e s e a r c h group:

Graeme Dewar, B r e t i s l a v Heinr ich , and Rick Baartman

whose b r a i n s I picked merc i l e s s ly . I am indebted

t o t h e Nat iona l Research Council of Canada f o r funds

which supported t h i s work.

TABLE OF CONTENTS

Page

vii LIST OF TABLES

viii LIST OF FIGURES

0. Introduction

1. Electromagnetic Waves and Spin Waves

2. The Allegation of Lieu and Alexandrakis

3. The Experiment

4. Conclusion

APPENDIX: Nonlinear Effects in Ferromagnetic Metals by 0. L. S. Lieu and G. C.-Alexandrakis

BIBLIOGRAPHY

v i i

LIST OF TABLES

Page

27 3.1 P r o p e r t i e s of t he Supermalloy and Nickel used i n t h i s work. €i2 = p c 2 / 2 ~ w i s t h e square of a s c a l i n g l eng th , where p i s t h e r e s i s t i v i t y i n esu. The demagnetizing f i e l d i n t h e specimen is given by H = 4710 M . D x s

v i i i

LIST OF FIGURES

Page

14 1.1 Propagat ion cons t an t f o r t h e EM wave a s a func t ion of magnetic f i e l d .

1.2 Enlargement of FMAR region of F igure 1.1. Damping has been reduced i n t h e lower graph.

1 .3 Propagat ion cons t an t f o r t h e a c t i v e s p i n wave a s a func t ion of magnetic f i e l d .

2 .1 The p reces s ion of t h e magnet izat ion.

2.2 An e l l i p t i c a l p reces s ion of t h e magnet izat ion d i sp l ay ing non- l inea r i t y i n t h e presence of a l a r g e e x c i t a t i o n .

2.3 Lieu and Alexandrakis ' t h e o r e t i c a l r e s u l t s f o r n i c k e l .

3 . l a F i e l d s i n s i d e an i d e a l resonant c a v i t y .

3 . lb C a v i t i e s and sample from t h e 24 GHz t ransmiss ion system.

3.2 Tracings of t he t ransmiss ion s i g n a l s observed f o r t he 40 u m t h i c k Supermalloy d i s c u s ing t h e p a r a l l e l - p a r a l l e l con f igu ra t ion ( r f and s t a t i c magnetic f i e l d s p a r a l l e l ) . Two orthogonal phases a r e shown; t h e s e d a t a were combined t o g ive t h e t ransmiss ion ampli tude vs . f i e l d shown i n Fugure 3.4. The maxima correspond t o a residuum of t h e usua l FMAR s i g n a l . Th i s r e s i d u a l FMAR s i g n a l corresponded t o a peak power of

-15 lo w a t t s f o r an i n c i d e n t power of 113 w a t t *

The bandwidth of t h e system was l i m i t e d by an output t i m e c o n s t a n t of 1.25 seconds.

3.3 Transmission ampli tude vs . magnetic f i e l d s t r e n g t h f o r t h e 40 um t h i c k Supermalloy d i s c u s ing t h e u s u a l p a r a l l e l - perpendicular FMAR conf igu ra t ion . The peak s i g n a l corresponded t o a power t ransmiss ion r a t i o of -91 db. It occurred a t 0.78 koe. The s o l i d curve was c a l c u l a t e d us ing t h e parameters l i s t e d i n Table 3.1 and a Landau-Lifshitz damping parameter of 1.15 x 10 Hz.

3.4 The two or thogonal t ransmiss ion phases of F igure 3.2 measured f o r t h e Supermalloy d i s c u s ing the p a r a l l e l - p a r a l l e l con f igu ra t ion have been combined t o g ive t ransmiss ion amplitude vs . magnetic f i e l d (O). These d a t a have been superposed on t h e FMAR t ransmiss ion s i g n a l of F igu re 3.3 (+). These t ransmiss ion curves have been normalized t o t h e same peak t ransmiss ion amplitude: t h e FMAR s i g n a l is approximately 600 t imes l a r g e r

in amplitude than the parallel-parallel signal.

36 3.5 Transmission amplitude vs. decreasing magnetic field for a 7.2 vm thick polycrystalline nickel foil. (x): The usual FMAR configuration having rf and static fields orthogonal. The peak transmitted power ratio was -110 db, and occurred at 1.2 koe. (+): The parallel-parallel configuration in which the rf and static fields were parallel. Both curves are plotted on the same scale.

Mi ritrovai per una selva oscura, Che la diritta via era smarrita.

In fe rno 1.1

Chapter 0

I n t r o d u c t i o n

Despi te t h e f a c t t h a t magnetism has been known s i n c e c l a s s i c a l t imes,

i t has not been u n t i l t h i s century t h a t a q u a n t i t a t i v e theory has e x i s t e d

f o r permanent magnets. This i s b e s t a s c r i b e d t o t h e f a c t t h a t t h e

mediat ing f o r c e is quantum mechanical i n o r i g i n .

Condensed m a t e r i a l s i n which a magnet iza t ion arises spontaneously

a r e c a l l e d ferromagnets . I f we t ake a s a model the no t ion t h a t a ferromagnet

is made up of e lementary magnets (avoiding t h e ques t ion of t h e source

of t h e f i e l d , momentarily), i t occurs t o u s t h a t a magnetic d i p o l e p r e s e n t s

an a l i g n i n g f o r c e t o nearby d i p o l e s and, t h e r e f o r e , we may e x p l a i n t h i s

macroscopic magnet izat ion i n pu re ly c l a s s i c a l t e r m s .

I n f a c t , however, we know t h i s n o t t o be p o s s i b l e . Of t h e va r ious

sources of magnetic d i p o l e moment, we know, a t t h e atomic s c a l e , t h a t t h a t

i n t r i n s i c t o t h e e l e c t r o n i s t h e s t r o n g e s t . The energy a s s o c i a t e d wi th the

alignment of neighboring e l e c t r o n s may be t h e r e f o r e compared wi th the energy

a s soc i a t ed wi th thermal e x c i t a t i o n s t o show t h a t no macroscopic alignment

i s p o s s i b l e above a few k e l v i n s .

In s t ead we know t h a t t h e f a m i l i a r room-temperature magnet izat ion

is quantum mechanical i n o r i g i n and has t h e s c a l e of a coulomb i n t e r a c t i o n .

Moreover, the f a c t t h a t t he e l e c t r o n has s p i n 4 ( i n t r i n s i c angular

momentum 8 ) impl ies t h a t i t s magnetic moment may be of no h ighe r 2

orde r than d i p o l e and t h a t t h i s moment be p a r a l l e l t o the s p i n . (See,

f o r i n s t a n c e , t he d e r i v a t i o n of (XIII.86) i n Messiah (1962).)

Although the o r i g i n of t he a l i g n i n g f o r c e i s known, i t s magnitude

i s n o t , s i n c e i t s e v a l u a t i o n involves an i n t e g r a l , t he "exchange"

i n t e g r a l which con ta ins c o n t r i b u t i o n s from every e l e c t r o n i n a macroscopic

specimen. (See, however, some t h e o r e t i c a l r e s u l t s by H i l l and Edwards

(1973) .) Thus, t h i s f o r c e cons t an t , the exchange cons t an t , l i e s i n

t he realm of t h e empi r i ca l .

From t h e c l a s s i c a l viewpoint , one may t ake t h e a l i g n i n g f i e l d as

a s o l e l y l o c a l magnetic d i p o l e f i e l d ( t h e Weiss f i e l d ) of l a r g e magnitude.

We can then make e s t i m a t e s of t he s t r e n g t h of t h i s f i e l d based on t h e

s u s c e p t i b i l i t y above t h e Curie temperature (based on the Curie-Weiss law)

and the s a t u r a t i o n magnet izat ion below the Curie p o i n t (based on a

Langevin f u n c t i o n ) . (See f o r i n s t ance , Chapter 16 , K i t t e l (1971) .)

Unfortunately, we know from quantum mechanics ( s ee , f o r example $10.11

Ziman (1972)) t h a t t h e thermal e x c i t a t i o n of a ferromagnet involves a

l a r g e frequency spectrum of exchange coupled e x c i t a t i o n s (spin-waves).

The s tudy of fe r romagnet ic resonance (FMR) was undertaken t o ga in

informat ion about t h e response of a ferromagnet t o monochromatic

r a d i a t i o n . I n p a r t i c u l a r , t he exchange cons t an t a t low (microwave)

f requencies i s of i n t e r e s t .

A magnet w i th an a s s o c i a t e d angular momentum executes a c i r c u l a r

precess ion when placed i n a cons t an t magnetic f i e l d . Although the e f f e c t

is more complicated when t h e magnet i s embedded i n a meta l ( d e t a i l s

t o be found i n t h e next c h a p t e r ) , t h e r e e x i s t s a p reces s iona l motion

i n ferromagnets having a c h a r a c t e r i s t i c f requency. This p reces s iona l mode

can be e x c i t e d by microwaves a s a resonance phenomenon, t he frequency

of which depends most s t r o n g l y on t h e s a t u r a t i o n magnet izat ion, t he app l i ed

e x t e r n a l f i e l d , and t h e e l e c t r o n ' s gyro-magnetic r a t i o .

The theory I s h a l l p re sen t i n t h e next chapter fo l lows t h a t of

Ament and Rado (1955) and Rado and Weertman (1959).

Since t h e p o s i t i o n of t he FMR l i n e does n o t depend s t r o n g l y on the

exchange and t h e width of t he l i n e may o r may n o t depend d i r e c t l y upon

exchange (depending on the r e l a t i v e magnitude of t h e s e v e r a l damping

mechanisms: The dependence i s weak i n N i and Supermalloy, s t r o n g i n Fe) , -

it i s d e s i r a b l e t o have a method f o r measuring exchange r e l i a b l y ,

independent of t h e o t h e r parameters of t h e m a t e r i a l . Lieu and Alexandrakis

(1975) claimed they had performed an experiment i n good agreement wi th

t h e i r theory which had a s t r o n g dependence on exchange. I n t h i s t h e s i s

I w i l l show t h e e r r o r i n t h e i r theory , I w i l l show t h a t t h e c o r r e c t theory

f o r t h e i r exper imenta l con f igu ra t ion does n o t depend on exchange, and

I w i l l de sc r ibe a similar experiment and compare t h e r e s u l t s I obta ined

with t h i s theory .

I n a n e x t e r n a l magnetic f i e l d , t he b e s t coupl ing t o t h e magnet izat ion

of a ferromagnet is obta ined i f microwave r a d i a t i o n impinges on t h e sample

polar ized s o t h a t t h e microwave magnetic f i e l d i s perpendicular t o t he

magnet izat ion. Lieu and Alexandrakis claimed, however, t h a t a ferromagnet

could be e x c i t e d by microwaves po la r i zed p a r a l l e l t o t h e magnet izat ion.

I w i l l show t h a t t h i s i s f a l s e (except a t very h igh power l e v e l s ) and

t h a t a ferromagnet ic meta l i n t h i s con f igu ra t ion responds c h i e f l y a s a metal

with t h e f a m i l i a r ( ~ = 1 ) electro-magnet ic f i e l d s i n i t s i n t e r i o r , and not

Consider the l i l i e s of the f i e ld , how they grow; They t o i l not, neither do they spin.

Matthew 6.28

Chapter 1

Electromagnet ic Waves and Spin Waves

Consider e lec t romagnet ic waves i n vacuum. There a r e two degenerate

eigenmodes of propagat ion f o r e lec t romagnet ic p lane waves i n a given

d i r e c t i o n . By degenera te , I mean t h a t a l l waves of t h e same frequency

have t h e same wavelength. Thus, one may choose a h o r i z o n t a l and v e r t i c a l

b a s i s o r a l e f t and right-hand c i r c u l a r b a s i s ( o r indeed among a continuum

of o the r bases) ad l i b .

I n ferromagnet ic meta ls t h e r e a r e four modes ( a s I w i l l show below),

which depend on t h e phys i ca l p r o p e r t i e s of t h e medium, two dependent

c h i e f l y on t h e conduc t iv i ty 0, and two-dependent c h i e f l y on t h e f e r ro -

magnetic exchange cons tan t A . The degeneracy i s removed.

To be more p r e c i s e : I s h a l l cons ider a meta l t o be a c l a s s i c a l

continuum wi th a s c a l a r ( i s o t r o p i c ) l o c a l conduc t iv i ty 0, such t h a t

+ J = 0T . Moreover, I s h a l l assume t h a t O i s l a r g e :

(Al l formulae a r e i n cgs u n i t s . )

I s h a l l cons ider a ferromagnet ic meta l t o have the above p r o p e r t i e s

+ i n a d d i t i o n t o these : It has a non-zero magnet iza t ion M which has

a f i xed amplitude M, ( c a l l e d the s a t u r a t i o n magnet iza t ion) , bu t no t

n e c e s s a r i l y a f i xed d i r e c t i o n , everywhere i n t h e meta l . I w i l l no t make

any mention of t h e temperature s i n c e i t makes only q u a n t i t a t i v e (not

q u a l i t a t i v e ) changes i n a few parameters . The theory is a p p l i c a b l e

a t any temperature low enough s o t h a t the ampli tude of t he magnet izat ion

is unperturbed by the app l i ed f i e l d .

This magnet izat ion i s mediated by a " r e s t o r i n g force" wi th a coupling

o r e l a s t i c cons t an t A c a l l e d the exchange f o r c e , which tends t o a l i g n

the magnet izat ion. I f we model t h e energy of t h e magnet izat ion a t

- t he o r i g i n

and expand

we ge t

With t h i s i n mind we t a k e t h e form of t he exchange torque t o be

which has the dimensions of a torque d e n s i t y g-. This form s e c 2 cm

of t he torque equa t ion i s due t o Herr ing and K i t t e l (1951). A i s

. t h e r e f o r e i n dynes . This exchange torque s t r o n g l y evokes sec

t he f a m i l i a r

It i s t h e r e f o r e tempting t o speak i n terms of an e f f e c t i v e exchange

4 4 4

i nduc t ion . However, s i n c e 6 and H d i f f e r by a q u a n t i t y p a r a l l e l t o M ,

we may a s e a s i l y speak i n terms of an exchange f i e l d as an induct ion:

- - z A I L C -

AI; VZV

The Laplacian has t h e e f f e c t of g iv ing t h e d i f f e r e n c e between a s o r t -

4 + of average va lue of M i n t he neighborhood and M i t s e l f . Thus, as long

a s A is p o s i t i v e , t he exchange f i e l d w i l l tend t o a l i g n t h e magnet iza t ion

p a r a l l e l t o t h e near-by magnet iza t ion . The magnet iza t ion i n common

ferromagnets a r i s e s c h i e f l y (but s e e Argyres and K i t t e l (1953) f o r

t he c o n t r i b u t i o n from t h e e l e c t r o n o r b i t a l motion) from t h e magnetic

d ipo le moment of t h e e l e c t r o n . This moment fo l lows t h e same r u l e s

and i s always i n t he same s t a t e as t h e e l e c t r o n ' s i n t r i n s i c angular

momentum, i t s quantum mechanical s p i n . Thus, we can d e f i n e a p o s i t i v e Y

such t h a t

4

where L i s t h e c l a s s i c a l angular momentum d e n s i t y due t o s p i n . Both

the angular momentum and t h e magnet iza t ion a r e a t p r e s e n t thought t o

be fundamental t o t he e l e c t r o n and d e s p i t e t he sugges t ive names, no c a u s a l

r e l a t i o n s h i p has been e s t a b l i s h e d between s p i n and magnetic moment.

Nonetheless, t he two terms a r e o f t e n used in te rchangeably , and, i n

p a r t i c u l a r , e x c i t a t i o n s which a r e coupled by the exchange f i e l d a r e c a l l e d

s p i n waves, t o d i s t i n g u i s h them from those coupled by t h e o rd ina ry

e lec t romagnet ic f i e l d . It should be emphasized t h a t t h e exchange f i e l d

i s quantum mechanical i n o r i g i n and i s much s t r o n g e r than t h e ord inary

magnetic coupl ing of near-by s p i n s . It a r i s e s from t h e P a u l i exc lus ion

of Fermions and i n f a c t t h e term "exchange" comes from t h a t con tex t .

I s h a l l d e a l w i th t h e s e geometr ies: Consider an i n f i n i t e s h e e t

of ferromagnet ic meta l w i th one s u r f a c e pas s ing through the o r i g i n . -

It w i l l be i n f i n i t e i n t h e x and z d i r e c t i o n s wi th f i n i t e o r i n f i n i t e

th ickness towards +y. An r f EM e x c i t a t i o n coming from t h e vacuum w i l l have

i t s magnetic f i e l d po la r i zed a long z . This e x c i t a t i o n w i l l be a

plane-wave d i r e c t e d a long y . A cons t an t magnetic f i e l d w i l l be app l i ed

e i t h e r a long x o r z .

sheet

Since t h e DC f i e l d i s always p a r a l l e l t o t he s l a b s u r f a c e , I w i l l

omit t he term " p a r a l l e l " a s used by most a u t h o r s from t h i s chapter

and use " p a r a l l e l " and "perpendicular" t o r e f e r only t o the r e l a t i v e

o r i e n t a t i o n of t h e r f and DC magnetic f i e l d s .

I w i l l f i r s t cons ider t h e perpendicular con f igu ra t ion .

Since we have a r e s t o r i n g torque (1.2) a c t i n g on an angular momentum

(1.4) we expec t p reces s iona l motion. I a s s e r t e d a t t he o u t s e t t h a t f o r

a given frequency t h e r e would be fou r non-degenerate propagat ion v e c t o r s .

1 For t h e approximation I adduce (1 .1) , Maxwell's equat ions a r e

Following Lieu and Alexandrakis , I w i l l use a propagator of t he form

(Please no te t h a t i s a wave-number i n t h i s work. Small l e t t e r s

w

1- TO s e e t h a t va=0, no te V*N= '%' d hl t hus ~7 = -2 =O. For plane d t

waves t h i s imp l i e s p =0.

w i l l be used f o r ampli tudes.) This y i e l d s ( f o r p lane waves)

-- - he, - C ( A ~ + 4"-)

-C

Since M i s of cons t an t l eng th and a l r e a d y o r i e n t e d a long x, any change

i n M,is second-order i n M, and Mz. We can , i n t h e l i m i t of smal l

s i g n a l s , t a k e m,=O. Thus we have

but on ly

where I have de f ined t h e s k i n depth

Equation (1.8) r e p r e s e n t s a n eigenmode of t h e medium uncoupled from

t h e magnet izat ion which w e w i l l t h e r e f o r e des igna te NM (non-magnetic).

To f i n d the s o l u t i o n s of (1.7) we need the torque equat ion r e l a t i n g

H and M, t h i s time con ta in ing both t h e exchange f i e l d (1.3) and t h e

r e a l f i e l d .

Magnetic e x c i t a t i o n s i n meta ls a r e c h i e f l y damped by coupl ing t o e l e c t r o n

motion ( sp in -o rb i t coupl ing) . Since t h e d e t a i l s of t h e damping do n o t

i n t e r e s t us , i t is convenient t o i n t roduce magnetic damping us ing a

phenomenological r e l a x a t i o n time T (Bloembergen damping). The i n t r o d u c t i o n

of damping i n t h i s p a r t i c u l a r form is merely t o keep accord wi th Lieu

and Alexandrakis . Using (1.4)

where I have a g a i n dropped second-order terms and used 6,=0. Combining

But from (1.7)

Thus 0 = A'

This equa t ion , be ing cub ic i n h2 con ta ins t h e remaining t h r e e e igenvalues .

We can r e a d i l y s e e t h a t on ly one e x i s t s i n t h e absence of exchange s i n c e ,

2 as A-0, t he equa t ion becomes l i n e a r i n h .

Let me then examine t h i s l i m i t i n g case , and, subsequent ly, the l i m i t

i n which s p i n waves dominate, t o i l l u m i n a t e t h i s i n i t i a l l y opaque equat ion .

I d e f i n e

The reason f o r my choice of phase f o r (1.15) i s t o make I' mostly

p o s i t i v e r e a l , the r e l a x a t i o n time usua l ly be ing long . I n t h e no-exchange

l i m i t , then,

This i s the so-ca l led EM wave, t he most i n t e r e s t i n g and exper imenta l ly

a c c e s s i b l e wave i n t he meta l . It has p o t e n t i a l l y two i n t e r e s t i n g po in t s :

where t h e denominator goes through (near ) zero and where t h e numerator does.

F igures 1.1 and 1.2 r e p r e s e n t t he EM s o l u t i o n of t he f u l l equat ion (1.13)

a s a f u n c t i o n of app l i ed f i e l d . Both t h e r e a l and imaginary p a r t s a r e

shown. It may be seen t h a t i f I' i s not l a r g e enough ( i . e . i f w is

no t ) wi th r e s p e c t t o M,, t h e numerator may n o t go through (near ) zero

and t h e second of t h e resonant p o i n t s w i l l be missing. The po in t

a t which t h e denominator approaches zero is c a l l e d ferromagnet ic resonance

(FMR). The p o i n t a t which t h e numerator approaches zero i s ferromagnet ic

ant i - resonance (FMAR). Although i t may seem c l e a r t h a t bo th of t hese

p o i n t s a r e of i n t e r e s t , t he importance of FMAR (indeed i t s ex i s t ence )

was not w e l l known u n t i l po in ted ou t by Heinr ich and Meshcheryakov

(1969, 1970). A t n f A R the damping i s l e a s t , t h e s k i n depth i s g r e a t e s t ,

and t h e r e is a consequent t ransmiss ion maximum. It i s common t o f i x

the frequency of t h e r a d i a t i o n and s lowly sweep t h e a p p l i e d magnetic

f i e l d . I n t h i s ca se t h e r e i s a s p e c i f i c f i e l d corresponding t o FMR

and ( i f a s u f f i c i e n t l y h igh frequency is chosen) another corresponding

t o FMAR. It goes wi thout say ing t h a t du r ing t ransmiss ion experiments ,

FMAR i s t h e impor tan t p o i n t . A p a t i e n t i n s p e c t i o n of F igures 1.1 and

1 . 2 ( a s o l u t i o n of t he f u l l s e c u l a r equa t ion (1.13)) r e v e a l s t h a t

Figure 1.1 Propagation constant for the EM wave as a function of magnetic field.

2 al though t h e r e i s simple l i n e a r behavior i n a t FMAR, t h e r e i s a d i p i n

t he r e a l (damping) p a r t of h . Moreover, s i n c e A appears i n an

exponen t i a l (1 .6 ) , the e f f e c t of a sma l l decrease i n t he r e a l (damping)

p a r t of h is g r e a t l y accentua ted i n any s i g n a l appear ing a t t he

back s u r f a c e of t h e s l a b .

Now l e t m e t u r n my a t t e n t i o n t o t h e s p i n waves. I n t u i t i v e l y , t hese

a r e t he s o l u t i o n s of t h e torque equa t ion (1.10) without r e f e rence t o the

consequent t o Maxwell's equa t ions (1 .7) , i. e . where t h e short-range

exchange f o r c e dominates. The e a s i e s t way t o do t h i s i s i n (1.11)

t o s e t t h e magnetic f i e l d h t o zero . This g ives

I f I d e f i n e

the s o l u t i o n of which i s

EM

Enlargement

Real FMAR

Figure 1.2 Enlargement of FMAR r eg ion of F igure 1.1. Damping has been reduced i n t h e lower graph.

One s e e s quick ly t h a t h2 is l i n e a r w i th f i e l d . One of t h e r o o t s i s

l a r g e and p o s i t i v e everywhere and t h e r e f o r e s t r o n g l y damped and

u n i n t e r e s t i n g . The o t h e r has a zero c r o s s i n g a t

i n t h e l i m i t of l a r g e T , t he r e l a x a t i o n time. Thus t h i s r o o t a l s o

has i n t e r e s t i n g behavior a t FMR ( see F igure 1 . 3 ) . It should be added,

however, t h a t t y p i c a l l y t h i s wave has a t h a t i s q u i t e l a r g e :

100 of t h e o rde r of - 6 . This causes i t t o be l a r g e l y unexci ted a t t h e

i n t e r f a c e , the h f o r vacuum being n e a r l y zero .

A l l t h r e e s o l u t i o n s a r e l a r g e l y unperturbed when they a r e combined

i n t o t h e f u l l s e c u l a r equat ion , a s may be seen from t h e f a c t t h a t ,

a l though t h e f i g u r e s a r e a c t u a l s o l u t i o n s of t h e f u l l equat ion (1.13),

they r e p r e s e n t q u i t e w e l l t h e behavior of t h e l i m i t i n g cases (1.16, 1 .18) .

The only p o i n t a t which mixing of s o l u t i o n s begins t o occur i s a t the

appl ied f i e l d corresponding t o FMR, when t h e spin-wave comes down

toward zero and t h e EM wave goes through a l o c a l i z e d maximum. However,

s i n c e our p re sen t i n t e r e s t i s i n FMAR, I w i l l n o t exp lo re t h e d e t a i l e d

s o l u t i o n of equat ion (1.13) any f u r t h e r .

2 'Spin Wave

Real -

Figure 1.3 Propaga t ion cons t an t for the a c t i v e s p i n wave as a func t ion of magnetic f i e l d .

Ye t i n thy dark s t r e e t s sh in i t h The everlas t ing l i g h t .

"0 Little Town" Bishop Phillips Brooks

Chapter 2

The Allegation of Lieu and Alexandrakis

I'd like to turn now to the precessional motion. Although I may

M. -

M

Y* -,

Figure 2.1 The precession of the magnetization.

t

have misled you whi le I was speaking of a magnet izat ion p reces s ing i n

an e x t e r n a l f i e l d , t h e r e i s no a priori reason f o r t h e excurs ion of the

magnet izat ion t o be c i r c u l a r . The complete torque equat ion (1.10)

con ta ins s e v e r a l terms, and the presence of boundaries i n t h e y-d i rec t ion

F igure 2.2 An e l l i p t i c a l p reces s ion of t he magnet izat ion d i s p l a y i n g non- l inea r i t y i n t he presence of a l a r g e e x c i t a t i o n .

1 with t h e consequent build-up of s u r f a c e magnetic charge i s an adequate

reason i n gene ra l f o r an e l l i p t i c a l excurs ion a s p i c tu red i n Figure 2.1.

I n t h i s f i g u r e and t h e n e x t , we look i n t h e d i r e c t i o n of +y towards t h e

sample s u r f a c e . The cons t an t magnetic f i e l d i s app l i ed a long x . Since

t h e magnet izat ion i s f i x e d i n ampli tude, i t becomes c l e a r t h a t t h i s excurs ion

can cause a second-order change i n Mx a s i n F igure 2.2, i. e . ,

Now wi thout s t r e t c h i n g your c r e d u l i t y too f a r , I b e l i e v e I can s t a t e ,

wi thout y e t r e s o r t i n g t o mathematics, t h a t t h i s sma l l change i n M, can

be used t o e x c i t e any e l l i p t i c a l (no t c i r c u l a r ) p r e c e s s i o n a l mode.

Consider t h e s t eady s t a t e . I f t h e r e i s an a l t e r n a t i n g magnetic f i e l d

i n t h e x -d i r ec t ion a t twice t h e p r e c e s s i o n a l f requency, i t w i l l c l e a r l y

t r a n s f e r energy t o a p reces s ing d i p o l e . Since t h e amount of energy

t r a n s f e r r e d ( A€= B,AM,) i t s e l f depends upon AM,, t h e energy t r a n s f e r

fo l lows an exponen t i a l growth o r decay law. Thus we s e e t h a t t h e r e is

a threshold f o r t h e e x c i t i n g f i e l d , below which the d i p o l e w i l l r e l a x

t o r e s t .

This coupling, where the e x c i t i n g f i e l d i s p a r a l l e l t o t he s t a t i c

f i e l d , i s what i s commonly c a l l e d p a r a l l e l pumping. I n due course ,

i t was observed i n an i n s u l a t o r (YIG) a t s u f f i c i e n t l y l a r g e r f f i e l d s

( - 1 oe) by Schlsmann e t a l . (1960). This f i e l d can be e a s i l y shown

1. I am speaking h e r e i n analogy wi th t h e e l e c t r o s t a t i c ca se . By 1

magnetic charge, I mean Q.F r - YX v.4

9 2 2 t o correspond t o 1.2 x 10 erg/cm -sec (120 wattlcm ) i n f r e e space .

We (1976) have c a l c u l a t e d t h e th re sho ld f o r t h e appearance of p a r a l l e l

12 2 pumping i n n i c k e l t o be ~ 8 5 oe corresponding t o 8.7 x 10 erg/cm -sec.

Lieu and Alexandrakis (1975, reproduced a s t h e Appendix) r epo r t ed

an experiment and theory reminiscent of p a r a l l e l pumping i n meta ls a t

much lower power l e v e l s . They p re sen t a theory which p r e d i c t s an

e x c i t a t i o n of t h e magnet iza t ion f o r low power l e v e l s i n t he p a r a l l e l

con f igu ra t ion . A s a func t ion of f i e l d , t he ampli tude they expec t i s

shown i n F igure 2.3. They observed a t ransmiss ion q u a l i t a t i v e l y s i m i l a r

t o t h i s . Thei r theory runs a s fo l lows .

-

Figure 2 . 3 Lieu and Alexandrakis ' t h e o r e t i c a l r e s u l t s f o r n i c k e l .

They s t a r t much a s I have by adducing Maxwell's equa t ions (1.5)

and the torque equat ion (1.10). They a l s o w r i t e t he component of t he

torque p a r a l l e l t o t h e magnet izat ion, a second-order term. These comprise

t h e i r equat ions (1 ) . They do n o t , however, use t h e l o n g i t u d i n a l torque

equat ion when d e r i v i n g t h e i r s o l u t i o n f o r t h e wave-number (2 ) . Hence,

the s o l u t i o n they g e t i s i d e n t i c a l t o (1.13) . They nex t t ake t h e

t roub le t o w r i t e t h e magnitude of t he l o n g i t u d i n a l d i s tu rbance i n terms

of t he ampli tudes of t he t h r e e t r a n s v e r s e ones. They next s t a t e t h a t they

w i l l use 10 boundary cond i t i ons . Forming equa t ions (5) a r e 6 of them,

while t h e o t h e r 4 a r e equa t ions (8). It is c r u c i a l t o r e a l i z e t h a t they

- so lve j u s t 10 boundary cond i t i ons .

Let me r e c a p i t u l a t e , b u t t h i s time speaking only of one s u r f a c e ,

so t h a t t h e number of wave ampli tudes and boundary cond i t i ons i s c u t

i n h a l f .

They s o l v e f o r t h e propagat ion v e c t o r s of wave-like d i s tu rbances

i n t h e magnet izat ion. Since they ignore t h e f i e l d s a t t h a t po in t ,

they miss t h e s o l u t i o n which l eaves t h e magnet iza t ion unexci ted , t h e

one I c a l l e d hM. Since one mode i s miss ing from t h e meta l , i t fo l lows t h a t , t o be

so luab le , t h e boundary va lue problem must be changed. Let me show t h a t

t h e problem a s o r i g i n a l l y s t a t e d r equ i r ed 6 cond i t i ons : Consider

a semi - in f in i t e sample s o t h a t we need d e a l on ly wi th t h e f i r s t su r f ace .

Consider t h a t t he gene ra l monochromatic wave i s i n c i d e n t upon i t ,

so t h a t an amplitude i s needed f o r each of t h e p o l a r i z a t i o n s . These

two ampli tudes a r e given. We must f i x t he ampli tudes of t he 2 r e f l e c t e d

waves and those of t h e 4 waves i n s i d e the meta l . We have immediately

t h e boundary cond i t i ons on E X , and hSy from c l a s s i c a l e lectrodynamics.

For t he remaining 2 cond i t i ons , we must t u r n t o Rado and Weertman (1959).

Their gene ra l t rea tment a l lows of t h r e e l i m i t i n g cases of which mx,y=O

i s a common choice ( K i t t e l ' s c o n d i t i o n ) . However, t h e form of t he

boundary cond i t i ons is n o t c r u c i a l t o my argument a t t h i s p o i n t .

What i s c r u c i a l i s t h a t t h e r e n e c e s s a r i l y be 2 cond i t i ons , of whatever

form.

This may be argued i n the' fo l lowing manner. Reca l l t h a t t h e magnet izat ion

has an a s s o c i a t e d angular momentum. The e x c i t a t i o n of a sample involves

t h e a p p l i c a t i o n of to rque by one i n t e r i o r e lementary magnet upon t h e

o t h e r s . The environment of t he s u r f a c e l a y e r magnets i s d i f f e r e n t from

those i n t he bulk . Thus, t he torque on t h e s e i s going t o be d i f f e r e n t from

the o t h e r s . I n gene ra l , t h e r e a r e t h r e e components of to rque , bu t s i n c e

4

MZ = MS a c t s a s a c o n s t r a i n t , on ly two cond i t i ons on M a r e necessary.

I have shown t h a t 6 boundary cond i t i ons e x i s t . I have s a i d t h a t Lieu

and Alexandrakis solved f o r only 5 wave ampl i tudes . To succeed a t t h i s ,

they ignored one of t h e boundary cond i t i ons , t o w i t , my=O(when d e r i v i n g

t h e i r equat ion (5) ) . That t h e i r problem y i e l d s a non-zero s o l u t i o n

a t a l l depends on these matching e r r o r s .

God said t o Abraham, " K i Z Z me your son. I f

Abraham said, "You must be Putting me on! I f

God said, "No!" Abe said, "Why? I f

God said, I'you can do what you Want, Abe, but the Next time you see me coming, You'd be t ter run!

"~ ighway 61" Bob Dylan

Chapter 3

The Experiment -

Lieu and Alexandrakis r e p o r t a t t h e same time a s they presented

t h e i r theory, r e s u l t s of t ransmiss ion experiments on i r o n and n i c k e l

f o i l s . These both have t h e proper ty t h a t t hey r i s e from zero amplitude

( i n t h e presence of no app l i ed magnetic f i e l d ) t o a f a i r l y f l a t p l a t e a u

above a few k i logauss . I n t h e experiments I performed, t h e r e a l s o appeared

a f l a t p l a t e a u above a few k i logauss . However, h e r e , t he t ransmiss ion

f e l l t o t h i s va lue r a t h e r than r o s e . Both of t hese observa t ions a r e

i n b a s i c agreement w i th the p r e d i c t i o n s made above of a non-magnetic mode,

bu t modulated nea r zero f i e l d by t h e motion of ferromagnet ic domain

boundaries .

EXPERIMENTAL DETAILS

The 24 GHz t ransmiss ion system used f o r t hese experiments was

developed and debugged by J . F. Cochran and B. Heinr ich t o whom I

g r a t e f u l l y acknowledge my d e b t . It is descr ibed more f u l l y elsewhere

(Cochran e t a l . (1977a)) . The specimens i n the form of t h i n d i s c s ( see

Table 3.1) were clamped between i d e n t i c a l c r i t i c a l l y coupled t r a n s m i t t e r

and r e c e i v e r c a v i t i e s , s o t h a t the specimen formed one end-wall of each

c a v i t y . (See F igure 3.1) The r f magnetic f i e l d s of t h e TE mode 10

Figure 3 . l a F i e l d s i n s i d e an i d e a l resonant c a v i t y .

C o u ~ l i n ~ Hole , -. , Teflon Tuning Rod -

-

Inc iden t + + Transmitted Radia t ion Radiat ion

-

T- \ Sample coppe; Diaphragm

Figure 3 . lb C a v i t i e s and sample from t h e 24 GHz t ransmiss ion system.

Table 3.1 P r o p e r t i e s of t h e S u p e m a l l o y and N i c k e l used i n t h i s work. 62 = p c 2 / 2 ~ u i s t h e s q u a r e of a s c a l i n g l e n g t h , where p i s t h e r e s i s t i v i t y i n e s u . The demagne t iz ing f i e l d i n t h e specimen i s g i v e n by H D = 4nD M . x S

Table 3 .1

D

ame t e cm

SPECIMEN D i

Nickel b

a Supermalloy 1.6

SPECIMEN

d

h ickness umc

p i n ' IJR cm a t 26OC

a Supermalloy

Nicke l b

59.8

7.8

a) Obtained from P e r f e c t i o n Mica Co. under t h e t r a d e name Conet ic AA. (740 Thomas Dr ive , Bensenvi l le , I l l . , 60106)

b) Goodfellow I n d u s t r i e s L td . , Ruxley Towers, Claygate-Esher, Sur rey , England. 99% pure , t h e ch i e f i m p u r i t i e s were Fe and Co.

u/y

(koeIe

8.14

7.85

Iris a r e a

mm2

2 .O

.4

c) Thicknesses were c a l c u l a t e d from t h e weight and a r e a of t h e specimens.

4 - S

(koele

7.38

6.06

d) Ca lcu l a t ed f o r t h e a p p r o p r i a t e e l l i p s o i d of r evo lu t ion .

e ) Measured i n our l a b o r a t o r y f o r s i m i l a r m a t e r i a l . See Cochran e t a l . , (1977a) and Dewar e t a l . (1977).

i n each c a v i t y were p a r a l l e l . The unloaded C) of each c a v i t y was approximately

1500. Approximately 300 mw of power was i n c i d e n t upon t h e t r a n s m i t t e r

-18 c a v i t y , and the r e c e i v e r had a s e n s i t i v i t y of 10 w a t t s us ing a

time cons t an t of 1.25 seconds. The c a v i t i e s were suspended between

the po le s of a Varian V-3800 15" e lec t romagnet . The magnet could be

r o t a t e d around a v e r t i c a l a x i s such t h a t the magnetic f i e l d remained

p a r a l l e l wi th the specimen p lane w i t h i n 1-2". A l l measurements were

c a r r i e d out on unannealed f o i l s a t room tempera ture , approximately 26OC.

Relevent c h a r a c t e r i s t i c s of t he two specimens used f o r t he p r e s e n t

experiments a r e l i s t e d i n Table 3.1. Discs were machined from the

r o l l e d f o i l by sandwiching the f o i l between f l a t p i eces of s tock ;

i n t h i s way i t was p o s s i b l e t o produce d i s c s wi thout bur red edges.

The n i c k e l d i s c was used without f u r t h e r p o l i s h i n g ; t h e Supermalloy

d i s c was e l e c t r o p o l i s h e d t o produce a sh iny s u r f a c e (Bloembergen (1950)) .

Transmission through each specimen was l i m i t e d t o an a r e a of a few mm 2

near i t s c e n t e r by means of an evaporated gold l a y e r 5-10 pm t h i c k .

It w a s hoped i n t h i s way t o minimize t h e non-homogeneity of t h e TE 10

r f d r i v i n g f i e l d , and t o minimize the e f f e c t of s p a t i a l inhomogeneities

i n t he s t a t i c magnet iza t ion caused by demagnetizing f i e l d s generated

a t t h e edges of t he d i s c .

Leakage of microwave r a d i a t i o n around t h e specimens was neg l igab ly

1 sma l l f o r a l l experiments r epo r t ed below . This could be checked by

sweeping t h e e x t e r n a l magnetic f i e l d through fe r romagnet ic resonance

1. I n t h e case of t he N i f o i l i t was found necessary t o s e a l t he c a v i t i e s u s ing conduct ing s i l v e r p a i n t a s descr ibed by Cochran e t a l . (1977a).

T

S

i n t h e o r i e n t a t i o n corresponding t o or thogonal r f and s t a t i c f i e l d s

( t h e p a r a l l e l - p e r p e n d i c u l a r c o n f i g u r a t i o n ) . It i s the exper ience

of our group t h a t f o r t h i c k specimens, such a s those used h e r e , any

leakage of r a d i a t i o n around the specimen r e s u l t s i n a f i e l d dependent

s i g n a l a t FMR (because of t h e modulation of t h e s u r f a c e impedence).

-15 A s m a l l f i e ld- independent background s i g n a l ( -10 , w a t t s ) i s

always p re sen t i n our appa ra tu s due t o d i r e c t leakage between the

t r a n s m i t t e r and r e c e i v e r wave-guide systems (p r imar i l y leakage through

t h e microwave swi tches used t o tune t h e c a v i t i e s and t o c a l i b r a t e t h e

r e c e i v e r s e n s i t i v i t y ) . This f i e l d independent background s i g n a l was

measured a t t h e e x t e r n a l f i e l d cor responding- to FMR wi th t he f i e l d

o r i e n t e d i n t h e pa ra l l e l -pe rpend icu l a r c o n f i g u r a t i o n . The background

s i g n a l was s u b t r a c t e d from t h e raw t r ansmis s ion d a t a , having due regard

f o r i t s phase. Transmission s i g n a l s hav ing t h i s background s u b t r a c t e d

are r e p o r t e d below.

RESULTS

I n our exper imenta l system, t ransmiss ion ampli tude i s measured

as a f u n c t i o n of magnetic f i e l d f o r two or thogonal phases . The d i g i t i z e d

d a t a f o r t h e two phases a r e then combined t o g ive t h e t r ansmis s ion

ampli tude and t h e f i e l d dependence of t h e phase. The r e s u l t s of an

experiment on the Supermalloy d i s c f o r t h e p a r a l l e l - p a r a l l e l c o n f i g u r a t i o n

are shown i n F igure 3.2. The l a r g e ampli tude v a r i a t i o n shown i n

F igure 3.2 was due t o a residuum of t h e FMAR s i g n a l : t h e t ransmiss ion

peak was approximately 600 t i m e s weaker than the FMAR s i g n a l observed

f o r t h e pa ra l l e l -pe rpend icu l a r c o n f i g u r a t i o n ( s ee F igure 3 .3 ) . The

Figu re 3 . 2 Trac ings of t h e t r ansmis s ion s i g n a l s observed f o r t h e 40 urn t h i c k Supermalloy d i s c u s ing t h e p a r a l l e l - p a r a l l e l con f igu ra t i on ( r f and s t a t i c magnetic f i e l d s p a r a l l e l ) . Two or thogonal phases a r e shown; t h e s e d a t a were combined t o g ive t h e t ransmiss ion ampli tude v s . f i e l d shown i n F igure 3.4. The maxima correspond t o a residuum of t he u s u a l F'MAR s i g n a l . This r e s i d u a l FMAR s i g n a l corresponded t o a peak power of

-15 approximately 10 w a t t s f o r an i n c i d e n t power of 113 wa t t . The bandwidth of t h e system w a s l i m i t e d by an ou tpu t t i m e cons t an t of 1.25 seconds.

Transmitted Amplitude (arb. units)

Figure 3.3 Transmission ampli tude vs . magnetic f i e l d s t r e n g t h f o r t h e 40 pm t h i c k Supermalloy d i s c u s ing t h e u s u a l pa ra l l e l -pe rpend icu l a r FMAR con f igu ra t i on . The peak s i g n a l corresponded t o a power t ransmiss ion r a t i o of -91 db. It occurred a t 0.78 koe. The s o l i d curve was c a l c u l a t e d u s ing t h e parameters l i s t e d i n Table 3.1 and a Landau- L i f s h i t z damping parameter of 1.15 x l o 8 Hz.

ampli tude of t h e n o i s e on the s i g n a l s shown i n Figure 3.2 corresponds

-18 approximately t o 10 w a t t s . Of p a r t i c u l a r i n t e r e s t a r e t h e t ransmiss ion

s p i k e s which occur red nea r zero f i e l d . These t r ansmis s ion s p i k e s were

bounded by approximately '15 oe on e i t h e r s i d e of ze ro f i e l d , and were

2 due t o t h e presence of domain boundaries i n t h e specimen . The va lue

15 oe f o r t h e d e p a r t u r e f i e l d ob ta ined from t h e d a t a of F igure 3.2

ag rees w e l l w i th t h e va lue 4"MS D~ = 14.5 oe (D is t h e x-demagnetizing x

f a c t o r ) given by elementary magne tos t a t i c arguments which a r e expected

t o be v a l i d f o r a s o f t magnetic m a t e r i a l such a s Supermalloy.

A p l o t of t r ansmis s ion ampli tude as a f u n c t i o n of app l i ed magnetic -

f i e l d f o r t h e Supermalloy d i s c i s shown i n Figure 3.3 f o r t h e u sua l

FMAR c o n f i g u r a t i o n , i. e., t h e r f and t h e s t a t i c f i e l d s were or thogonal .

The peak occurred a t a f i e l d Ha = 0.78 koe; t h i s i s c l o s e t o t he va lue

W H a = - - 4nM, = 0.76 koe p red i c t ed by theory , n e g l e c t i n g magnetic damping

Y

The maximum shown i n F igure 3.3 corresponded t o a t r ansmis s ion ampli tude

r a t i o of 2.8 x o r a power r a t i o of -91 db. This power r a t i o i s

t h e q u o t i e n t of t he power reaching t h e microwave r e c e i v e r and t h e

power i n c i d e n t upon t h e t r a n s m i t t e r c a v i t y : i t t h e r e f o r e depends

2. W e can be c e r t a i n t h a t t h e t r ansmis s ion s p i k e s near zero f i e l d f o r t h e p a r a l l e l - p a r a l l e l c o n f i g u r a t i o n , as w e l l a s t h e t ransmiss ion d i p shown i n F igure 3.3, were due t o domain w a l l s because both of t h e s e t r ansmis s ion f e a t u r e s d i sp l ayed a small bu t s i g n i f i c a n t h y s t e r e s i s when t h e e x t e r n a l f i e l d was cyc led from l a r g e p o s i t i v e t o nega t ive va lues . The peak p o s i t i o n s were s h i f t e d approximately 5 oe . A t r ansmis s ion peak of s i m i l a r o r i g i n h a s been observed i n Metglas 2826 f o r t h e p a r a l l e l - p a r a l l e l c o n f i g u r a t i o n (Cochran e t a l . , 1977b). Metglas 2826 (Fe40Ni40P14B6, a n amorphous

ferromagnet) is a t r a d e name of t h e A l l i e d Chemical Corporat ion, Morristown, N. J .

upon geomet r ica l f a c t o r s such a s specimen a p e r t u r e s a s w e l l a s upon

t h e c a v i t y q u a l i t y f a c t o r s . The measured power r a t i o agreed , w i th in

a f a c t o r of 2, w i th t h a t c a l c u l a t e d by Cochran e t a l . (1977a) t ak ing

the specimen geometry i n t o account .

The f i e l d v a r i a t i o n of t ransmiss ion ampli tude f o r both t he p a r a l l e l -

p a r a l l e l and pa ra l l e l -pe rpend icu l a r c o n f i g u r a t i o n s a r e shown superposed

and normalized t o t he same peak ampli tude i n F igure 3 . 4 . It should

be borne i n mind t h a t t h e peak ampli tudes d i f f e r i n s t r e n g t h by approximately

600. Apart from the ze ro f i e l d s i g n a l s , t h e s e t r ansmis s ion curves

a r e s u f f i c i e n t l y s i m i l a r a s t o l eave no doubt t h a t t h e p a r a l l e l -

p a r a l l e l s i g n a l was j u s t t h e u s u a l FMAR s i g n a l e x c i t e d by a s m a l l

component of t h e r f f i e l d or thogonal t o t h e magnet iza t ion . This

conc lus ion i s r e i n f o r c e d by t h e obse rva t ion t h a t when t h e a c t i v e a r ea

2 2 of t h e specimen was i nc reased from 2 mm t o 10 mm t h e r a t i o of t h e

3 pa ra l l e l -pe rpend icu l a r t o p a r a l l e l - p a r a l l e l ampli tude decreased t o 25.

We do no t know why t h e p a r a l l e l - p a r a l l e l t ransmiss ion peak i s a d i s t o r t e d

ve r s ion of t h e FMAR peak - a s i m i l a r d i s t o r t i o n and s h i f t toward

h igher f i e l d s has a l s o been observed f o r specimens of t h e amorphous

meta l a l l o y Metglas 2826. It may be an e f f e c t due t o s u r f a c e roughness.

Small s u r f a c e i r r e g u l a r i t i e s would be expected t o g ive r eg ions w i t h i n

t h e skin-depth where t h e l o c a l magnet iza t ion would d e v i a t e from

3 . It should be noted t h a t t h e d i f f e r e n c e between t h e s e r a t i o s i s probably due i n p a r t t o t h e d i f f e r e n c e i n s h i e l d i n g techniques . I n t h e second ca se , i n s t e a d of t h e evaporated gold l a y e r , t h e sample was sh i e lded by a f r ee - s t and ing copper diaphragm. Thus t h e r e was probably some propaga t ion of microwaves p a r a l l e l t o t h e sample s u r f a c e behind t h e diaphragm. However, w e know independent ly t h a t t h e r e was no leakage .

the d i r e c t i o n of t he bulk s a t u r a t i o n magnet iza t ion . A l t e r n a t i v e l y ,

i t may be t h a t t h e presence of a t h i c k (0 .3 mm) copper diaphragm

(See Figure 3.1) caused a smal l imhomogeneity i n t he r f magnetic

f i e l d . Although I observed no th ing i n t he t ransmiss ion through t h e 40ym

Supermalloy specimen f o r t he p a r a l l e l - p a r a l l e l con f igu ra t ion which

resembled the t ransmiss ion s i g n a l s r epo r t ed f o r i r o n and n i c k e l by

Lieu and Alexandrakis (1975), i t could be argued a ) t h a t t h e i r observa t ions

were s p e c i f i c a l l y concerned wi th i r o n and n i c k e l , and t h a t b) t h e

r a t i o of specimen th i ckness t o skin-depth f o r t h e Supermalloy specimen was

too l a r g e t o r e v e a l t h e i r e f f e c t . I n o r d e r - t o d u p l i c a t e t h e i r experiment

a s c l o s e l y a s p o s s i b l e I t h e r e f o r e a l s o measured t ransmiss ion through

a p o l y c r y s t a l l i n e n i c k e l f o i l f o r which d l6 = 7.9, a va lue c l o s e t o

4 t h a t used by Lieu and Alexandrakis (d/6 = 6.8) . The r e s u l t s of t he

experiment a r e shown i n F igure 3.5. Transmission f o r t he p a r a l l e l -

perpendicular con f igu ra t ion , t h e usua l FMAR conf igu ra t ion , is shown

on t h e same s c a l e a s t h e t ransmiss ion f o r t h e p a r a l l e l - p a r a l l e l

con f igu ra t ion . The peak s i g n a l a t FMAR corresponded t o an amplitude

- 6 r a t i o of 3 . 3 x 10 , o r a power r a t i o of -110 db. The f i e l d independent

p l a t e a u i n t he p a r a l l e l - p a r a l l e l con f igu ra t ion corresponded t o an

amplitude r a t i o of 5.8 x An es t ima te of t h i s t ransmiss ion r a t i o

f o r a non-magnetic meta l having t h e DC r e s i s t i v i t y of n i c k e l , and

2 4 . Lieu and Alexandrakis used 6 = (pc /4nw)" = 3 . 2 um; they c a l c u l a t e d

the skin-depth us ing p = 19 pRcm, a va lue approximately 24 t imes l a r g e r than the DC r e s i s t i v i t y of n i c k e l .

Transmitted Amplitude (arb. units)

Figure 3.4 The two orthogonal transmission phases of Figure 3.2 measured for the Supermalloy disc using the parallel-parallel configuration have been combined to give transmission amplitude vs. magnetic field (a). These data have been superposed on the FMAR transmission signal of Figure 3.3 (+). These transmission curves have been normalized to the same peak transmission amplitude: the FMAR signal is approximately 600 times larger in amplitude than the parallel-parallel signal.

Transmitted Amplitude (arb. units)

Figure 3.5 Transmission ampli tude v s . dec reas ing magnetic f i e l d f o r a 7.2 pm t h i c k p o l y c r y s t a l l i n e n i c k e l f o i l . (x): The usua l FMAR c o n f i g u r a t i o n having r f and s t a t i c f i e l d s or thogonal . The peak t r ansmi t t ed power was -110 db, and occur red a t 1.2 koe. (+): The p a r a l l e l - p a r a l l e l c o n f i g u r a t i o n i n which t h e r f and s t a t i c f i e l d s were p a r a l l e l . Both curves a r e p l o t t e d on t h e same s c a l e .

t ak ing the c a v i t y q u a l i t y f a c t o r s and a c t i v e a r e a of t he specimens

-7 i n t o account a s descr ibed by Cochran e t a l . (1977a), gave 5 x 10 . This agreement between t h e c a l c u l a t e d and observed s i g n a l s t r e n g t h

i s s t r o n g evidence t h a t the bulk of t h e s i g n a l observed f o r t he p a r a l l e l -

p a r a l l e l con f igu ra t ion was due t o t he o rd ina ry skin-depth mode which

i s expected t o be e x c i t e d i n the metal when the r f and s t a t i c f i e l d s

a r e p a r a l l e l . The t ransmiss ion peak near zero f i e l d was due t o f e r r o -

magnetic domain boundaries a s evidenced by t h e f a c t t h a t i t e x h i b i t e d

h y s t e r e s i s . The displacement between t ransmiss ion peaks due t o

h y s t e r e s i s was approximately 100 oe. Domain w a l l e f f e c t s occurred over

a r e l a t i v e l y l a r g e f i e l d i n t e r v a l i n t h e n i c k e l specimen because of

magneto-crystal l ine an i so t ropy coupled wi th t h e p o l y c r y s t a l l i n e

n a t u r e of t he specimen.

DISCUSSION

Apart from a f i e l d i n t e r v a l near zero , our observa t ions on the

t ransmiss ion through a n i c k e l f o i l f o r t he p a r a l l e l - p a r a l l e l con f igu ra t ion

a r e i n agreement wi th the observa t ions r epo r t ed by Lieu and Alexandrakis,

namely a f i e l d independent s i g n a l . The ampli tude of t h i s s i g n a l

f o r bo th s e t s of d a t a i s c o n s i s t e n t wi th the hypothes is t h a t i t i s

simply due t o t h e o rd ina ry e lec t romagnet ic wave which we expect t o be

e x c i t e d i n a ferromagnet ic meta l when the r f and s t a t i c magnetic f i e l d s

a r e p a r a l l e l . We d id not observe t h i s background s i g n a l i n our

experiment on Supermalloy because the specimen was twice a s t h i c k

i n t e r m s of skin-depths a s was t h e n i c k e l specimen (See Table 3.1).

This was s u f f i c i e n t t o cause the e lec t romagnet ic wave t o be a t t e n u a t e d

t o an unobservably smal l l e v e l .

Near zero f i e l d we observed an enhanced t ransmiss ion (See the

lower curve of Figure 3.5) which was c l e a r l y due t o ferromagnet ic domain

w a l l motion because the peak occurred a t d i f f e r e n t f i e l d s depending upon

t h e h i s t o r y of t he magnetic f i e l d c y c l e . Lieu and Alexandrakis observed

a n a t t e n u a t e d t ransmiss ion near zero f i e l d . Poss ib ly t h i s a t t e n u a t i o n

near ze ro f i e l d i s an e f f e c t due t o t h e motion of s u p e r f i c i a l domain

w a l l s . It can be argued t h a t t he presence of domain w a l l s w i t h i n t h e

specimen may e i t h e r d iminish o r i n c r e a s e t ransmiss ion depending

upon t h e i r l o c a t i o n and o r i e n t a t i o n . Those confined t o the sample

s u r f a c e would tend t o s h i e l d i t s i n t e r i o r from microwaves. Consider,

however, domain w a l l s which extend through t h e specimen. It is t h e

l a t t e r type of domain w a l l which i s r e spons ib l e f o r enhanced t ransmiss ion

near ze ro f i e l d . Thus the d i f f e r e n t r e s u l t s r epo r t ed by our two groups

could be due t o d i f f e r e n c e s i n samples which favored the dominance

of one s o r t of domain w a l l over t h e o t h e r .

A more d e t a i l e d i n v e s t i g a t i o n of t h e ze ro - f i e ld t ransmiss ion

anomaly i s c u r r e n t l y underway i n our l a b o r a t o r y us ing m a t e r i a l s f o r

which the domain s t r u c t u r e can be we l l - cha rac t e r i zed .

Can you picture yourseZf, So ZimitZess and free, Desperately i n need Of some str2anger ' s hand?

"The End" The Doors

Chapter 4

Conclusion

I have pointed ou t t h e e r r o r i n t h e theory of Lieu and Alexandrakis.

I n t h e r e a l m of s a t u r a t e d magnet iza t ion , t h e i r d a t a can be explained -

i n terms of t h e l i n e a r t heo ry . Where t h e i r d a t a d i f f e r s q u a l i t a t i v e l y from

mine ( i . e . i n t he reg ion below s a t u r a t i o n ) t h e d i f f e r e n c e may w e l l

be due t o d i f f e r e n c e s i n sample p repa ra t ion which inf luenced domain

w a l l motion.

The e f f e c t they observe i s very probably t h e non-magnetic ( f i e l d

independent) mode (of equat ion 1 . 8 ) , bu t modulated by domain w a l l motion

at f i e l d s f o r which t h e sample is d iv ided i n t o magnetic domains.

I must add t h a t , except ing only t h e i r exper ience , I know of no reason

t o expec t t he conduc t iv i ty a t 24 GHz t o be d i f f e r e n t ( e s p e c i a l l y s o

markedly) from the DC conduc t iv i ty . I have f i t t e d t h e d a t a i n Figure 3 . 3

without r e s o r t t o t h i s . Their use of a conduc t iv i ty d i f f e r i n g by a

f a c t o r of 2% ( a s noted i n foo tno te 4 of t h e last chap te r ) from the DC

va lue seems excess ive .

APPENDIX

NONLIKEAR EFFECTS 1N FERROI,?AGNETIC lIETALSf

O.L.S. L i e u and G.C. A lexand rak i s Phys i cs Department, U n i v e r s i t y o f Miami

Co ra l Gables, F l o r i d a 33124

ABSTRACT

A n o n l i n e a r t h e o r y o f fe r romagnet ic t r a n s m i s s i o n resonance i n t h e case i n wh ich t h e s t a t i c and microwave f i e l d s a r e m u t u a l l y p a r a l l e l and p a r a l l e l t o t h e sanp le s u r f a c e i s o u t l i n e d . I t s p r e d i c t i o n s a r e compared t o expe r imen ta l r e s u l t s o b t a i n e d f o r i r o n and n i c k e l . The t h e o r y p r e d i c t s s t r o n g subharmonic g e n e r a t i o n o f p o s s i - b l e p r a c t i c a l use.

THEORY

The e f f e c t s d e s c r i b e d he re a r e s t r i c t l y ferromag- n e t i c . E x p e r i m e n t a l l y t h e y a r e n o t observed i n para- magnet ic meta ls . ' Assume t h e sample i s - i n f o i l fo rm o f t h i c k n e s s d and i s made t o form t h e common w a l l o f two i d e n t i c a l microwave c a v i t i e s . One c a v i t y i s used t o s t o r e t h e i n c i d e n t microwaves o f f requency wg and t h e o t h e r t o c o l l e c t t h e power t r a n s m i t t e d th rough t h e sample. Take t h e sample t o l i e on t h e xz-plane. The microwaves propagate a long t h e y - a x i s . Bo th t h e ap 1 i e d s t a t i c and microwave f i e l d s a r e a long t h e z-ax is.'^ !

The Bloch-Bloembergen and Maxwell equa t i ons f o r t h i s prob lem a r e w r i t t e n as:

9 = ~ ( H ~ H , - M I H ~ ) - %4 V'M, -$ \

Here = (Hx,Hy ,Ha+Hz), fi = (Mx,My,Ms+Mz) where Ha i s t h e a p p l i e d s t a t i c f i e l d , MS i s t h e s a t u r a t i o n magnet- i z a t i o n , Hx,y and Mx a r e t h e h i g h f requency compo- nen ts of t h e f ~ e l d s . &her symbols have t h e i r s tanda rd meanings.

Assume t h e p l a n e wave s o l u t i o n s ~ ~ , ~ = h : ' ~ e - ( ~ y + i w t ) s

M, y=mXsY e-(*ytiut) f o r t h e t r ansve rse components o f Eqa. (1). We o b t a i n t h e s e c u l a r e q u a t i o n f o r t h e p r o p a g a t i o n c o n s t a n t x by u s i n g t h e t r a n s v e r s e component equa t i ons f rom Eqs. ( I ) , t o be

X~ + a2h4 + alX2 + a. = 0 (2 ) where

Equa t i on (2 ) i s c u b i c i n A ? , t h e . r o o t s a r e + A 1 , rA , 1 h 3 wh ich can be expressed i n terms o f ao, al, p2 a n a l y t - i c a l l y . They g i v e t h e wave p ropasa t i on a long +y and -9.

We now w r i t e ~ , = ~ ( a ~ i * i ~ + B jeA~ l ) iiwt (j:9,2,3) (3

and s i m i l a r J e x p r e s s i o n s f o r Mx,My,Hy where A j and 0 . a r e t h e amp l i t udes t o be determined by boundary cond i - t i o n s . Us ing these exp ress ions o f t h e t r a n s v e r s e f i e l d s i n t h e l o n g i t u d i n a l components o f (1) we s o l v e f o r HZ. The s o l u t i o n i s

b,,~),~i'"t H,=X h L

(4 1 where

N o t i c e t h a t t h e f i e l d t r a n s m i t t e d a long t h e z - a x i s

i s a t t h e a p p l i e d f r equency wg= 2w whereas t h e one t r a n s m i t t e d a long t h e x - a x i s has t h e f requency w = w0/2.

We now b e g i n t o s o l v e t h e boundary va lue prob lem f o r t h e t r a n s m i t t e d f i e l d s . The boundary c o n d i t i o n s a r e (a ) t h e t a n g e n t i a l components o f fi and a r e con t i nuous ac ross t h e sample s u r f a c e s a t y=O and y = d . (b) K i t t e l ' s boundary c o n d i t i o n f o r t h e t r a n s v e r s e magne t i za t i on , i . e . Mx=O a t y=o,d. We a r range t h e boundary c o n d i t i o n s i n two groups. The f i r s t group which c o n t a i n s t h e boundary c o n d i t i o n s on Hx, EZ and Mx, i s w r i t t e n i n m a t r i x form as

where

Hr and H tx a r e t h e r e f l e c t e d and t r a n s m i t t e d amp l i - t u i e s a1 ong t h e x -ax i s .

From (51 we have

( ; ~ = - G ' ( ~ , A ~ P ~ - \ 2 ~ , ~ 3 ) ( : ) + +E~G,-LIG-'[<) .

where

$Q)=' '% ) K.=%, Hrf i s t h e i n c i d e n t microwave

amplit:d:d~, and Htz a re the r e f l g c t e d and t ransmi t ted amp1 i tudes arong the z-axis.

The HtZ and Htx a r e obta ined from (8) t o be

HtE= H~~ c ~ - ' ~ ' 'Z (9

H,, = 12H,l P'*:' xtt1 (10)

where - -

H,= ~~i A e'(K-"tZ) z where A i s the a m p l i f i c a t i o n f a c t o r and 4 i s an a r b i t r a r y phase in t roduced by the apparatus.

The power t ransmi t ted along the z-ax is i s then p ropor t iona l t o 12(+) 1 w h i l e the one t r a n s m i t t e d along the x-ax is i s p ropor t iona l t o I X ( + ) 1 2 .

DISSCUSSION

I n F ig . l ( a ) and l ( d ) we have p l o t t e d the experimental t ransmi t ted amplitudes along the z- d i r e c t i o n f o r a 18pm t h i c k Fe f o i l and a 10 pm t h i c k N i sample, a t room temperature. The f r e - quency o f HZt which i s the same as the e x c i t a t i o n frequency i s c lose t o u = 9.2 GHz. The theore t - i c a l r e s u l t s f o r lHztl ?or Fe and Ni correspond- i n g l y a r e p l o t t e d i n F i g . l ( b ) and l ( e ) . The c a l c u l a t e d lHxt values f o r Fe and Ni are shown i n F i g . l ( c ) and 1 I f ) . As was i n d i c a t e d above the Hxt f i e l d has ha1 f the app l ied frequency.

( f ) thoo .

0 2 4 6 8 I 0 He+ it^

F I G . 1 (a) expe r imen ta l , ( b ) , ( c ) t h e o r e t i c a l r e s u l t s f o r Fe. (d) expe r imen ta l , ( e ) , ( f ) t h e o r e t i c a l r e s u l t s f o r N i . Parameter va lues a r e g i v e n i n t e x t .

The parameter va lues used f o r t h e Fe p l o t s a re : MS = 1707 G, g = 2.05, T = 1.0 x 10- O sec, T1 = 1 x 10 - ' 0 sec, 6 = 5 um and A = 2.5 x ergs/cm. F o r N i we used: M = 485, g = 2.2, T = 1.0 x 1 0 - l o , T = 1.0 x 6 = 3.2urn and A = .9 x l o b 6 ergsfcni. The va lues we used f o r 6 which i n o u r t h e o r y e n t e r s f o r t h e f requency v0/2 a r e 502 l a r g e r than t h e i r suggested va lues f rom s t a t i c r e s i s t i v i t y data . T h i s i s i n accordance w i t h o u r p a s t exper ience i n me ta l s . The A v a l u e f o r N i used he re i s s l i g h t l y l a r g e r t h a n t h e pub- l i s h e d v a l u e 3 o f A = .75 x ergs/cm. The A v a l u e f o r Fe i s t h e same as t h e p u b l i s h e d va lue .3

The c a l c u l a t e d va lues f o r t h e I H Z t l ' S a r e about one o r d e r o f magnitude s m a l l e r than t h e i r expe r imen ta l va lues. The (HX t l 1s a r e about two o r d e r o f magnitude s t r o n g e r than t h e I H z t l ' s . T h i s i s g e n e r a l l y c o n s i s t e n t w i t h t h e f a c t t h a t t h e H x t ' s have a l a r g e r s k i n dep th t han t h e H ? ~ ' s .

I n summary, what appears t o be t h e case i n t h e phenomena desc r i bed here i s t h a t a t l o w s t a t i c f i e l d va lues most o f t h e i n c i d e n t power a long t h e z - a x i s , i s pmped i n t o t h e HXty compo- nen ts t h rough e x c i t a t i o n o f t r a n s v e r s e s p i n waves, r e m i n i s c e n t o f p a r a l l e l pumping i n i n s u l a t o r s .4

However a t h i g h s t a t i c f i e l d va lues most o f t h e power i s r e t a i n e d by t h e H Z t component a f t e r a l l o w i n g f o r normal a t t e n u a t i o n i n t h e m e t a l .

We have observed t h e H t r a n s m i s s i o n resonance phenomenon i n p r a c t i c a f f y a1 1 f e r romagne t i c me ta l s . Bes ides 6,on which a l l microwave p ropaga t i ons i n me ta l s have a s t r o n g dependence, t h e c a l c u l a t e d H Z t ' s a r e v e r y s t r o n g l y dependent on A c o n t r a r y t o o t h e r l i n e a r b u l k resonance effect^.^ Thus t h e n o n l i n e a r i t y i s p redominan t l y dependant on A. We p l a n t o f i t t h e exper imenta l r e s u l t s w i t h t h e t h e o r y t o t r y t o e s t a b l i s h t h e tempera tu re dependence o f t h e exchange cons tan t . The p re - d i c t e d subharmonic gene ra t i on m i g h t have p r a c t i c a l use i n s p e c i a l cases. T h i n f i l m s w i l l have t o be used, a l t h o u g h even then t h i s method c o u l d n o t compete w i t h t h e p resen t harmonic g e n e r a t i n g d iodes i n most cases. We a r e p r e s e n t l y p r e p a r i n g t o d e t e c t t h e lHxt 1 and assess i t s usefu lness.

We w i s h t o thank O r . M.A. Hue r ta f o r h e l p f u l sugges t i ons r e g a r d i n g t h e c a l c u l a t i o n s .

REFERENCES

*Work suppo r ted by t h e N a t i o n a l Sc ience Foundat ion .

G.C. A lexand rak i s , 0. Horan, T.R. Carver, and C.N. Manicopoulos, Phys. Rev. L e t t . 25, 1758 (1970). . --- - , - O.L.S. L i e u and G.C. A lexand rak i s and M.A. Huer ta , t o 5e pub l i shed . B.E. A r g y l e , S . Charap, and E.W. Pugh, Phys. Rev. 132, 2051 (1963); F. Kef fer , Handbuck d e r P h y s i k f V I I I /2, 30 (Spr in,ger-Ver lag, B e r l i n , 1966); C. He r r i ng , Rado and Suhl , Elagnetism I V , -

Academic Press, New York, 1966, p. 357. F.R. Morgen tha le r , J. App l . Phys. ?_, 955 (1960); E. Schlomann, J.J. Green, and U. M i l ano , J. App l . Phys. 3J. 3865 (1960); C. K i t t e l , Quantum Theory o f S o l i d s , (W i l ey , New York, 1964),p. 69; M. S p a r k s , - i - e r r o m a p e t i c - ~ e h x a t i o n Theory, (McGraw- H i l l , New York, 1 9 6 4 r c o n t a i n s an e x t e n s i v e b i b l i o g r a p h y on n o n l i n e a r e f f e c t s i n i n s u l a t o r s . 0. Horan, G.C. A lexand rak i s , and C.N. Manicopoulos, Phys. Rev. L e t t . 25. 246 (1970). ,

BIBLIOGRAPHY

Alexandrak i s , G. C . , Horan, 0.. Carver , T. R . , and Manicopoulos, C . N. 1970. Phys. Rev. L e t t . 25, 1758.

Ament, W. S. and Rado, G. T. 1955. Phys. Rev. - 97, 1558.

Argyres , P. and K i t t e l , C . 1953. Acta M e t . - 1, 241.

Bloembergen, N . 1950. Phys. Rev. 3, 572.

Cochran, J. F., H e i n r i c h , B . , and Dewar, G. 1 9 7 7 ( a ) . To be pub l i shed i n t h e Canadian J o u r n a l o f P h y s i c s .

Cochran, J . F., H e i n r i c h , B . , and B a a r t m n , R . 1977(5) . P roceed ings I C X 76. To be pub l i shed i n Phys ica B ( S e t h ) .

Dewar, G. , H e i n r i c h , B . , and Cochran, J . F. 1977. To be p u b l i s h s d i n t h e Canadian J o u r n a l of P h y s i c s .

H e i n r i c h , B. and Meshcheryakov, V. F. 1969. ZhETF P i s . Red. - 9, 618. JETP L e t t . - 9 , 378 (1969) .

H e i n r i c h , B. and Meshcheryakov, V. F. 1970. ZhETF - 59, 424. S o v i e t Phys. JETP - 32, 232 (1971) .

Her r ing , C . and K i t t e l , C . 1951. Phys. Rev. - 81 , 869.

H i l l , D . J . and Edwards, D. M. 1973. J . Phys. F (GB) - 3, L162.

K i t t e l , C . 1971. I n t r o d u c t i o n t o S o l i d S t a t e P h y s i c s . 4 t h e d . (Wiley, New York) .

Kurn, A. , Cochran, J . F., and H i e n r i c h , B. 1976. Can. J. Phys. 54, 1083.

Lieu, ~ 0 . L. S. , and Alexandrak i s , G. C . 1975. AIP Conf. Proc. 24, 512, e d . by C . D. Graham, G. H . Lander, and J . J. Rhyne (AIP, New York) .

Messiah, A l b e r t 1962. Quantum Mechanics. ( N o r t h - ~ o l l a n d , ~ m s t e r d a m ) .

Rado, G. T. and Weertman, J . R. 1959. J . Phys. Chem. S o l i d s 2, 315.

Schlomann, E., Green, J . J . , and Milano, U. 1960. J. Appl. Phys. 31, 386s. -

Ziman, J . M. 1972. P r i n c i p l e s of t h e Theory of S o l i d s . 2nd e d . (Cambridge U n i v e r s i t y P r e s s , London).